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% Author       : 焱铭
% Date         : 2023-07-04 20:57:50 +0800
% LastEditTime : 2023-07-05 13:51:06 +0800
% Github       : https://github.com/YanMing-lxb/
% FilePath     : \SCI-LaTeX-Submission-Process-for-Elsevier\1-Manuscript-CN\Section\Section3.tex
% Description  : 
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\section{Numerical method}

In this study, the following assumptions were made to simplify the numerical model: 

\begin{enumerate}[1.] % 可选类型 a) (i) Step 1.
    \item Newtonian fluid flow and steady laminar flow are used, and the fluid follows the Hagen-Poiseuille equation.
    \item The walls of the channel are rigid.
    \item Neglecting the effects of interaction forces, viscous heat, surface tension, and radiative heat transfer.
\end{enumerate}

\subsection{Governing equations and boundary conditions}

The governing equations are as follow:

Continuity equation:
\begin{equation}
    \nabla \cdot \left(\rho_f \vec{u}\right)=0
\end{equation}

Momentum equation:
\begin{equation}
    \vec{u} \cdot \nabla\left(\rho_f \vec{u}\right)=-\nabla p+\nabla \cdot\left(\mu_f \nabla \vec{u}\right)
\end{equation}

Energy equation for the fluid domain:
\begin{equation}
    \vec{u} \cdot \nabla\left(\rho_f C_{f} T_f\right)=\nabla \cdot\left(k_f \nabla T_f\right)
\end{equation}

Energy conservation equation for the solid domain:
\begin{equation}
    \nabla\left(k_s \nabla T_s\right)=0
\end{equation}
\textcolor{red}{}


The density ($\rho_{f}$), specific heat ($C_{f}$), thermal conductivity ($k_{f}$), and viscosity ($\mu_{f}$) of deionized water are correlated with the temperature as shown below:

\begin{figure*}[htb] % 将长公式放入figure* 环境中进行跨栏显示
    \begin{align}
        \rho_{f}(T)= & 999.9+9.561 \times 10^{-2} T-1.013 \times 10^{-2} T^{2}+8.459 \times 10^{-5} T^{3}-3.496 \times 10^{-7} T^{4}                      \\ \notag\\
        C_{f}(T)=    & 4217-3.452 T+1.155 \times 10^{-1} T^{2}-1.862 \times 10^{-3} T^{3}+1.538 \times10^{-5}T^{4}-4.850 \times 10^{-8} T^{5}             \\ \notag\\
        k_{f}(T)=    & 5.698 \times 10^{-1}+1.772 \times 10^{-3} T-4.870 \times 10^{-6} T^{2}-2.915 \times10^{-8} T^{3}+1.094 \times 10^{-10} T^{4}       \\ \notag\\
        \mu_{f}(T)=  & 1.750 \times 10^{-3}-5.558 \times 10^{-5} T+1.172 \times 10^{-6} T^{2}-1.579 \times10^{-8} T^{3}+1.169 \times 10^{-10} T^{4}\notag \\
                     & -3.535 \times 10^{-13} T^{5}
    \end{align}
\end{figure*}


where T is the temperature ($^{\circ}C$)


\subsection{Data reduction}


The performance of the four microchannel heat sinks was evaluated by their thermal resistance, Mean Absolute Temperature Deviation (MATD), and pressure drop.
Thermal resistance is defined as follows \cite{Ansari.Jeong_2021}:
\begin{equation}
    R_{th}=\frac{T_{c.max }-T_{in.max }}{Q_{tot}}
\end{equation}
$Q_{tot}$ is the chip's total heat produced, represented as:
\begin{equation}
    Q_{tot}=SV_{c}
\end{equation}

\subsection{Grid independence}